In [1]}, Petitjean et al. proposed to replace
multiple sequences with an average sequence \(\hat{T}=\arg \min _{\hat{T}}\lbrace DTW(T_{i},\hat{T}) \rbrace _{i=1}^{N}\)  using the DTW distance. But it does not
have a contribution on improving the alignment of DTW algorithm. Keogh and
Pazzani [2]} proposed to replace the coordinator
distances by the derivative distance (DDTW). Therefore, points with similar
changing trend are matched. Jeong et al. [3]} proposed
to weight the match of two points by the length of stretch such that the
long shift is penalized (WDTW). A combination of WDTW and DDTW is also considered (WDDTW). Batista et al. [4]}
proposed a similar penalty-based distance. That method additionally
considers a coefficient on top of the DTW distance in order to avoid small
distance between sequences of different complexity (CID-DTW). Like DTW,
DDTW, WDTW and CID-DTW still have the limitation in considering the
sequential nature of data. Other studies focus on improving the efficiency
of DTW [5]} but do not improve the alignment
results of DTW.
